Probability density function problem

[Nikitin]

Homework Statement

Let the probability density function##f(x) = (3/4) \cdot (1-x^2)## if x is between -1 and 1, and let ##f(x)=0## otherwise.

What is the probability of ##P(X \leq 0.8 | X>0.5)##?

Homework Equations

The Attempt at a Solution

I assume I have to rewrite the p.d.f. into a joint probability density function so I can use bayes' rule?

And what does the large "##X##" mean in probabilities? Often my texts uses large X-es instead of small. Is the large X for the actual event, while the small x is just a function variable?

 

[LCKurtz]

Homework Statement

Let the probability density function##f(x) = (3/4) \cdot (1-x^2)## if x is between -1 and 1, and let ##f(x)=0## otherwise.

What is the probability of ##P(X \leq 0.8 | X>0.5)##?

Homework Equations

The Attempt at a Solution

I assume I have to rewrite the p.d.f. into a joint probability density function so I can use bayes' rule?

And what does the large "##X##" mean in probabilities? Often my texts uses large X-es instead of small. Is the large X for the actual event, while the small x is just a function variable?

##X## denotes the random variable whose density function is ##f(x)##. So the probability that ##X## falls between ##a## and ##b## is$$
P(a\le X \le b) = \int_a^b f(x)~dx$$You should be able to use that formula along with the definition of conditional probability to solve your problem.

 

[Nikitin]

Could you put me on the right track? I'm not sure how to use conditional probability on this, as I don't know the probability for the intersection between "##X \leq 0.8##" and "##X>0.5)##".

 

[LCKurtz]

Could you put me on the right track? I'm not sure how to use conditional probability on this, as I don't know the probability for the intersection between "##X \leq 0.8##" and "##X>0.5)##".

What values of ##X## satisfy both inequalities?

 

[Ray Vickson]

Could you put me on the right track? I'm not sure how to use conditional probability on this, as I don't know the probability for the intersection between "##X \leq 0.8##" and "##X>0.5)##".

Draw a number line for X. Sketch the two regions X > 0.5 and X ≤ 0.8. What part of the line is the region common to both these larger regions? Using the given density, how would you compute the probability of that common region?

I urge you to struggle with this if necessary; asking for too much help on such problems hinders your learning and will not be good for you in the long run at exam time. If you cannot get it in 2 minutes, don't give up. If you need two hours, take two hours.

 

[Nikitin]

What values of ##X## satisfy both inequalities?

oh god how foolish of me. Thank you! I understand it all now.

Draw a number line for X. Sketch the two regions X > 0.5 and X ≤ 0.8. What part of the line is the region common to both these larger regions? Using the given density, how would you compute the probability of that common region?

I urge you to struggle with this if necessary; asking for too much help on such problems hinders your learning and will not be good for you in the long run at exam time. If you cannot get it in 2 minutes, don't give up. If you need two hours, take two hours.

I got it now. I was too confused to see what the intersection of the two events actually was (it was really stupid of me to not get it immediately). thanks for the advice!